Geodesic
What do Abelian categories form?
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 1, pp. 1-45

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Given two finitely presentable Abelian categories $A$ and $B$, we outline a construction of an Abelian category of functors from $A$ to $B$, which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of $A$ and $B$. The construction is absolute, so it makes it possible to recover not only Hochschild cohomology but also Mac Lane cohomology. Bibliography: 29 titles.
Keywords: Abelian category, stable category, 2-category, Hochschild cohomology, Mac Lane cohomology. 
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D. B. Kaledin. What do Abelian categories form?. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 1, pp. 1-45. http://geodesic.mathdoc.fr/item/RM_2022_77_1_a0/