# Questions tagged [hom-functor]

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### Lifting of a morphism in a hom-functor when hom-sets are empty

I'm having trouble in unserstanding a hom-functor. Suppose we have a hom-functor $Hom(X,$_$)$ for some Category $\mathcal{C}$. Suppose further that the hom-sets Hom$_{Set}(X,A)$ and Hom$_{Set}(X,B)$ ...
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### Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$

I'm trying to solve this exercise: So we are given that for all objects $B$, there is a natural isomorphism $$H_A(B)\simeq H_{A'}(B)$$ Let's write this isomorphism as $f\mapsto \bar f$. The ...
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### What is the name of this identity relating the monoidal products of finite sets and finite-dimensional vector spaces?

Note: This question is almost certainly a duplicate. Since I don't know the terminology involved, I couldn't find the original question. If someone can find the original question and link to it, then ...
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### There is a unique map $P \to \text{Hom}(A, \prod_i C_i)$ whenever $P \xrightarrow{p_i} \text{Hom}(A, C_i)$?

I'm trying to prove directly that $\text{Hom}(A, \prod_i C_i) \simeq \prod_i \text{Hom}(A, C_i)$ whenever $\prod_{i\in I} C_i$ is a product of a family of objects $C_i$ in a category $B$, where $A$ is ...
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