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Homelimit of a functor
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limit of a functor
Let $G$ be a functor from categories $\mathcal{I}$ to $\mathcal{C}$. A limit of $G$ is a pair $(L,\tau)$ where
1. $L:\mathcal{I}\to\mathcal{C}$ is a constant functor,
2. $\tau:L\to G$ is a natural transformation,
such that it is universal among all pairs satisfying (1) and (2). In other words, for any pair $(M,\varphi)$, where $M:\mathcal{I}\to\mathcal{C}$ is a constant functor, and $\varphi:M\to G$ is a natural transformation, there is a unique natural transformation $\phi:M\to L$ with the following commutative diagram:
$\xymatrix@C+=64pt{M\ar[dr]^{{\varphi}}\ar@{.>}[dd]_{{\phi}}\\ &G\\ L\ar[ur]_{{\tau}}}$
Since we may identify a constant functor with its value, say an object $A$, in $\mathcal{C}$, a limit of $G$ may be viewed as an object $A$ in $\mathcal{C}$, together with a collection of morphisms $A\to G(I)$, or $G_{I}$, for each object $I$ in $\mathcal{I}$ such that
$\displaystyle\mbox{if }I\to J\mbox{ is a morphism, then }A\to G_{I}\to G_{J}=A% \to G_{J}.$  $\displaystyle\qquad(*)$ 
Furthermore, if another object $B$ in $\mathcal{C}$ satisfies $(*)$, then there is a unique morphism $B\to A$ such that
$B\to A\to G_{I}=B\to G_{I}$ 
for all objects $I$ in $\mathcal{I}$.
Remarks.

A limit of a functor may or may not exist. If it exists, then it is unique up to natural equivalence. In other words, we may call it the limit of $G$.

In the literature, the limit of $G$ is variously known as the inverse limit, left limit, projective limit, root, or left root of $G$. and is generally written $\liminv G$.

The category $\mathcal{I}$ above is usually called the index category, and the functor $G$ a diagram in $\mathcal{C}$.

The most common index categories are finite categories, discrete categories, partially ordered categories, and more specifically, directed categories.

On the other hand, it turns out that if, in a category, if equalizers of any pairs of morphisms, and arbitrary products of any collections of objects exist, then it can be shown that every functor into this category has a limit.
By reversing all the arrows in $\mathcal{C}$, we arrive at the dual concept of a limit, that of a colimit. Precisely, the colimit of $G:\mathcal{I}\to\mathcal{C}$ is a pair $(R,\tau)$ where $R:\mathcal{I}\to\mathcal{C}$ is a constant functor and $\tau:G\to R$ is a natural transformation, such that for any pair $(M,\varphi)$ of constant functor $M:\mathcal{I}\to\mathcal{C}$ and natural transformation $\varphi:G\to M$, there is a unique natural transformation $\phi:R\to M$ such that the following diagram
$\xymatrix@C+=64pt{&R\ar@{.>}[dd]^{{\phi}}\\ G\ar[ur]^{{\tau}}\ar[dr]_{{\varphi}}\\ &M}$
The colimit is also known as the direct limit, right limit, inductive limit, coroot, or right root, and is written $\limdir$.
Examples. Some common examples of limits (inverse limits) are products, terminal objects, pullbacks, equalizers, kernels, and kernel pairs. These examples can be readily verified. Let us verify that a terminal object is a limit:
Let $G$ be the empty functor (from the empty category) into an arbitrary category $\mathcal{C}$. So the limit of $G$ is just an object $C$ in $\mathcal{C}$, and that’s it, as there are no objects in the empty category, there are no morphisms from $C$ in the limit of $G$. If $A$ is any object in $\mathcal{C}$, then there is a unique morphism $A\to C$, and that’s it. But this means that $C$ is just a terminal object of $\mathcal{C}$.
Please see the verification of some of these examples in the attachments below.
Some examples of colimits (direct limits) are coproducts, initial objects, pushouts, coequalizers, cokernels, and cokernel pairs.
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