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It is easy to define the tensor product as being the object that represents the bilinear maps functor, but to prove that tensor products exist requires something extra. If you have free abelian groups …

For each object $c$ in $\mathcal{C}$, let $c^* : [\mathcal{C}, \mathcal{A}] \to \mathcal{A}$ be evaluation at $c$. It is an exact functor, so if a left adjoint $c_! : \mathcal{A} \to [\mathcal{C}, \ma …

Observe that $\bigoplus_I : \textbf{Ab}^I \to \textbf{Ab}$ is a conservative exact functor: it is right exact by general nonsense, it preserves monomorphisms (because e.g. $\bigoplus_{i \in I} A_i$ is …

Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the struct …

Given a topological space or locale $X$ and an open $j : U \hookrightarrow X$ with closed complement $i : K \hookrightarrow X$, the inverse image functor $\langle i^*, j^* \rangle : \textbf{Sh} (X) \t …