# Riehl's “Category Theory in Context” - Exercise 3.4.i

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?

• I think the natural isomorphism should be $\mathsf{Cone}(X,F)\cong$Hom$(X,\lim F)$. – Matematleta Dec 28 '18 at 15:54

## 1 Answer

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.