I am afraid questions might be trivial and well-known, well, nevervethless. I'll ask it.

Consider category of vector spaces. There are naturally defined functors to itself : take vector space and consider its symmetric or antisymmetric powers, more general things can be constructed - which is called Schur functors (as far as I understand) they are indexed by Young diagrams. (They are defined as certain subspaces in tensor product $V^{\otimes n}$, for example if we take $V^{\otimes 3}$ then there are symmetric tensors, antisymmetric and there is certain complement to them - that is one of the Schur functors).

So we get many functors $F_{diagr}$. Functors can be composed. Are there any relations between composition of functors ? E.g. $F_{d1} F_{d2} = F_{d3} F_{d4}$ or whatever ?

Also functors on the catetory of vectors spaces can be added - so we have some algebra. What are the relations in this algebra ?