# Questions tagged [hom-functor]

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### Homomorphism into a union of $R$-algebras.

All rings/algebras appearing in this question are assumed to be commutative with unity and noetherian. Let $R$ be a ring, let $A, B$ be $R$-algebras, and let $(B_i)_{i \in I}$ be a family of sub-$R$-...
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### What is a clear, elegant proof that $\text{Hom}(\cdot, Y)$ is a right-exact functor in a category of modules?

If you search the site for this proof, you will find duplicates, however they are hard to understand. In other words they brush by the most critical points of the proof as if they were not worth ...
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### Hom is left exact, preserves limit

I'm trying to understand this sentence (From https://en.m.wikipedia.org/wiki/Exact_functor) "A covariant (not necessarily additive) functor is left exact if and only if it turns finite limits into ...
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### What does “$\text{Hom}(\space, G)$takes values in Groups” mean?

From Rotman's Algebraic Topology: When we say that $\text{Hom}(\space , G)$ takes values in Groups, then it follows, of course, that $\text{Hom}(X, G)$ is a group for every object $X$ and $g^*$ ...
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This is from Categories & Sheaves by Kashiwara & Schapira. Let $C$ be a category, $f : X \to Y$ a morphism in $C$. Assume that for each $W \in C$, the morphism $\text{Hom}_C(W, X) \... 2answers 97 views ### Yoneda Lemma (again) Let$C^{\wedge} = \text{Fct}(C^{\text{op}}, \text{Set})$. Let$\text{h}_C : C \to C^{\wedge}, \ X \mapsto \text{Hom}_C(\cdot, X)$. Then the Yoneda lemma is: For$A \in C^{\wedge}$and$X \in C$,$...
Here in the definition of weighted colimit, it seems to me that the r.h.s. $C(W⋅F,c)≅Set^{J^{op}}(W,C(F−,c))$ has a wrong direction for the application of the Yoneda lemma.In fact, the Yoneda lemma ...
I'm using the definition of fibre product from the Stacks Project. How do I show that $h_{x \times_y z}(w) = h_{x}(w) \times_{h_y(w)} h_z(w)$ ? Is equality correct or should it be isomorphism? I'm ...