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Let $\mathsf{I}$ be a small category, let $\mathsf{C}$ be a locally small category and let $F\colon\mathsf{I\to C}$ be a functor. Emily Riehl in her book "Category Theory in Context" defines a limit of a $F$ as a represenation of a specific functor.

$\mathsf{Cone}(-,F)\colon\mathsf{C^{op}}\to\mathsf{Set}$ sends each object $X$ of $\mathsf{C}$ to the set $\mathsf{Cone}(X,F)$ of all cones from $X$ to $F$ (that is, to that set $\mathsf{Hom}_{[\mathsf{I},\mathsf{C}]}(X,F)$ where $X$ is a constant functor) and sends each morphism $f\colon X\to Y$ of $\mathsf{C}$ to the functor $\mathsf{Cone}(f,F)\colon\mathsf{Cone}(Y,F)\to\mathsf{Cone}(X,F)$ so that for any cone $\lambda\colon Y\to F$ and for any $i \in \mathsf{I}$ we have $\mathsf{Cone}(f,F)(\lambda)_i = \lambda_i\circ f$.

A limit of $F$ then is an object of $\mathsf{C}$ which represents $\mathsf{Cone}(-,F)$.

It is known that a limit of any small set-valued functor $F\colon\mathsf{I}\to\mathsf{Set}$ is essentially $\{ (\lambda_i)_{i \in \mathsf{Ob(I)}} \in \prod_{i \in \mathsf{Ob(I)}}F(i) \mid $for any morphism $f\colon i\to j$ of $\mathsf{I}$ we have $\lambda_j = F(f)(\lambda_i) \}$

Let $X \in \mathsf{C}$. Consider the composition functor $\mathsf{Hom_C}(X,-)F\colon\mathsf{I}\to\mathsf{Set}$. From what we've said above it can be easily seen that a limit of this functor is the set $\mathsf{Cone}(X,F)$ of all cones from $X$ to $\mathcal{D}$, hence $\lim_{\mathsf{I}}(\mathsf{Hom_C}(X,-)F) \cong \mathsf{Cone}(X,F)$. Exercise $3.4.$i. asks to prove to this isomorphism is natural. But I'm not sure that does it mean for a morphism to be natural in this context. Of course, it has something to do with natural transformations, but what exactly?

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  • $\begingroup$ I think the natural isomorphism should be $\mathsf{Cone}(X,F)\cong $Hom$(X,\lim F)$. $\endgroup$ – Matematleta Dec 28 '18 at 15:54
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A natural transformation (isomorphism) is a morphism (isomorphism) between functors, so, first of all, you have to understand which functors are on the both sides of the natural isomorphism. There is $Cone(-,F)\colon C^{op}\to Set$ on the right side, because you are asked to prove naturality in $X$. The functor on the left side must have the same "type" $C^{op}\to Set$. An action on objects of this functor is given by formula $$X \mapsto \lim_i Hom_C (X, F-),$$ hence it remains to give an action on morphism. Given $f\colon Y \to X$ you have to define an arrow $$\lim_i Hom_C (f, F-)\colon\lim_i Hom_C (X, F-)\to\lim_i Hom_C (Y, F-).$$ It can be done using the universal property. The fact that for every $i\in Ob I$ $Hom_C(-,F(i))$ is contravariant in the first argument will also help you.

Thereafter you will be able to check the naturality of the isomorphism in the usual way.

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