I am trying to prove the next results,
Let $F: {}_{R}\mathfrak{M}\longrightarrow {}_{S}\mathfrak{M}$ a covariant funtor (between $R$-modules and $S$-modules categories). Suposse that such functor is left exact and it preserves direct limits. Show it preserves inverse limits.
and the "dual" assertion,
Let $F: {}_{R}\mathfrak{M}\longrightarrow {}_{S}\mathfrak{M}$ a covariant functor. Suposse that such functor is left exact and it turns coproducts into products. Show it turns direct limits into inverse limits.
I'm not able to use morita's theorems, so I was trying to give a direct proof of this, but I don't fin any conection between direct limits and inverse limits in this case, in the same case, I was trying to do something like this. Any hint?