# Questions tagged [hom-functor]

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### Weighted limits

I have a very trivial question about the page 80 here: how this shape of $W$ $$W:2\to \mathbf{Set}$$ with $$\ast\sqcup\ast\to \ast$$ implies that the components of $$W\Rightarrow\cal{M}(m,f)$$ are ...
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### $(X^Y)^Z\cong X^{(Y+Z)}$ or $(X^Y)^Z\cong X^{(Y\times Z)}$?

$\DeclareMathOperator\Hom{Hom}$I have the following exercise in my class of Category Theory: Prove that $\text{Hom}(Z,\Hom(Y,X))\cong \Hom(Y*Z, X)$ but I am not sure what $*$ is. I think that $*$ ...
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1 vote
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### $F$ left adjoint to $G \iff F,G$ define a functor from $\textbf{Arr}(\textbf{X}\times\textbf{A}) \to 2\times 1$ square CDs in $\textbf{Set}$?

Let $\textbf{A, X}$ be categories and $F:\textbf{X} \to \textbf{A}$ and $G: \textbf{A} \to \textbf{X}$. Then there is a map that takes an object in $\text{Arr}(\textbf{X}\times\textbf{A})$ (the arrow ...
• 19.5k
1 vote
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### How do you derive the adjoint's "naturality" condition as seen in MacLane & Moerdijk's book "Sheaves in Geometry and Logic"?

It's tag (7) as pictured below. I also included the definition of adjoint that they use. I know that by definition of adjunction (using the natural homset isomorphism), we have two naturality ...
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### Examples of not-full functors between two categories in both directions?

A functor F:C→D from a category C to a category D is said to be full if, for each pair of objects, x,y ∈ C, the function, F:C(x,y)→D(F(x),F(y)) between hom sets is surjective. What I am unsure about ...
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• 1,401
1 vote
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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-$R$-...
• 931
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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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