An initial object in a category $\mathcal{C}$ is an object $\emptyset$ such that for all objects $x \,\in\, \mathcal{C}$, there is a unique morphism $\varnothing \xrightarrow{\exists !} x$ with source $\varnothing$. and target $x$.
An initial object, if it exists, is unique up to unique isomorphism, so that we may speak of the initial object.
When it exists, the initial object is the colimit over the empty diagram.
Initial objects are also called coterminal, and (rarely, though): coterminators, universal initial, co-universal, or simply universal.
An initial object $\varnothing$ is called a strict initial object if all morphisms $x \xrightarrow{\;} \varnothing$ into it are isomorphisms.
Initial objects are the dual concept to terminal objects: an initial object in $C$ is the same as a terminal object in the opposite category $C^{op}$.
An object that is both initial and terminal is called a zero object.
An initial object in a poset is a bottom element.
Likewise, the empty category is an initial object in Cat, the empty space is an initial object in Top, and so on.
The trivial group is the initial object (in fact, the zero object) of Grp and Ab.
An initial object in a category of central extensions of a given algebraic object is called a universal central extension.
Let $\mathcal{C}$ be a category.
The following are equivalent:
$\mathcal{C}$ has a terminal object;
the unique functor $\mathcal{C} \to \ast$ to the terminal category has a right adjoint
Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.
Dually, the following are equivalent:
$\mathcal{C}$ has an initial object;
the unique functor $\mathcal{C} \to \ast$ to the terminal category has a left adjoint
Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.
Since the unique hom-set in the terminal category is the singleton, the hom-isomorphism characterizing the adjoint functors is directly the universal property of an initial object in $\mathcal{C}$
or of a terminal object
respectively.
By definition, an initial object is equipped with a universal cocone under the unique functor $\emptyset\to C$ from the empty category. On the other hand, if $I$ is initial, the unique morphisms $!: I \to x$ form a cone over the identity functor, i.e. a natural transformation $\Delta I \to Id_C$ from the constant functor at the initial object to the identity functor. In fact this is almost another characterization of an initial object (e.g. MacLane, p. 229-230):
Suppose $I\in C$ is an object equipped with a natural transformation $p:\Delta I \to Id_C$ such that $p_I = 1_I : I\to I$. Then $I$ is an initial object of $C$.
Obviously $I$ has at least one morphism to every other object $X\in C$, namely $p_X$, so it suffices to show that any $f:I\to X$ must be equal to $p_X$. But the naturality of $p$ implies that $\Id_C(f) \circ p_I = p_X \circ \Delta_I(f)$, and since $p_I = 1_I$ this is to say $f \circ 1_I = p_X \circ 1_I$, i.e. $f=p_X$ as desired.
An object $I$ in a category $C$ is initial iff $I$ is the limit of the identity functor $Id_C$.
If $I$ is initial, then there is a cone $(!_X: I \to X)_{X \in Ob(C)}$ from $I$ to $Id_C$. If $(p_X: A \to X)_{X \in Ob(C)}$ is any cone from $A$ to $Id_C$, then $p_X = f \circ p_Y$ for any $f:Y\to X$, and so in particular $p_X = !_X \circ p_I$. Since this is true for any $X$, $p_I: A \to I$ defines a morphism of cones, and it is the unique morphism of cones since if $q$ is any morphism of cones, then $p_I = !_I \circ q = 1_I \circ q = q$ (using that $!_I = 1_I$ by initiality). Thus $(!_X: I \to X)_{X \in Ob(C)}$ is the limit cone.
Conversely, if $(p_X: L \to X)_{X \in Ob(C)}$ is a limit cone for $Id_C$, then $f\circ p_Y = p_X$ for any $f:Y\to X$, and so in particular $p_X \circ p_L = p_X$ for all $X$. This means that both $p_L: L \to L$ and $1_L: L \to L$ define morphisms of cones; since the limit cone is the terminal cone, we infer $p_L = 1_L$. Then by Lemma we conclude $L$ is initial.
(relevance for adjoint functor theorem)
Theorem is actually a key of entry into the general adjoint functor theorem. Showing that a functor $G: C \to D$ has a left adjoint is tantamount to showing that each functor $D(d, G-)$ is representable, i.e., that the comma category $d \downarrow G$ has an initial object $(c, \theta: d \to G c)$ (see at adjoint functor, this prop.). This is the limit of the identity functor, but typically this is the limit over a large diagram whose existence is not guaranteed. The point of a solution set condition is to replace this with a small diagram which is cofinal in the large diagram.
Textbook accounts:
Francis Borceux, Section 2.3 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)