# The functor category Funct(R-Mod, S-Mod)

Can you name some properties of the functor category Funct(R-Mod,S-Mod), where R,S are associative rings with unit?

EDIT: I am sorry for the lack of precision of my (first in MO) question. I was thinking in a sort of Yoneda's Lemma for functors $F\colon R\mbox{-Mod}\to S\mbox{-Mod}$, maybe using tensorizing functors $h_U$ to describe $Nat(h_U,F)$.

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I think it might that your question has been downvoted because it comes off as vague. But I for instance am very ignorant about this category and only recently started reading about Morita theory. So answers would be nice :) –  Yosemite Sam Jan 28 '12 at 0:52
mathoverflow.net/howtoask - what properties? Is it very like a whale? –  Yemon Choi Jan 28 '12 at 1:30
Here is a definite question you might ask: One knows their is a "set"(?) map $ring(S,R)\rightarrow Fun(R-mod,S-mod)$. Other than elements in the image of this map, what else is there? What is a concrete example something not in this image? What are some conditions on $R$ and $S$ for this map to be an isomorphism? Answers to these questions might shed some light on the properties of the category. –  Spice the Bird Jan 28 '12 at 3:49
In Morita theory, you should consider the category of R-S bimodules. Tensoring with a bimodule gives all colimit preserving functors and so in particular all equivalences. Bimodules form a nice abelian category. –  Benjamin Steinberg Jan 28 '12 at 4:27
-1 since still there is no precise question at all. –  Martin Brandenburg Jan 28 '12 at 10:05