I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ as $\mathrm{Hom}(X,M)$ for some $X$. Do you know if we can? I would be happy also if you can give me only a reference.
(Here I'm talking of modules over commutative rings, we can also suppose that the $N_j$'s and $M$ are finitely generated, we can also add some other conditions if you wish, maybe noetherianity).
I also read that $\mathrm{Hom}(M,\lim\limits_{\longleftarrow}N_j)\cong\lim\limits_{\longleftarrow}\mathrm{Hom}(M,N_j)$. I was wondering if under some hypothesis $\mathrm{Hom}(M,\lim\limits_{\longrightarrow}N_j)\cong \lim\limits_{\longrightarrow}\mathrm{Hom}(M,N_j)$. I would be happy if you can help me even in only one of those questions.
EDIT: what about if $M$ is not even finitely generated?
EDIT: Suppose that the transition maps in $\{N_j\}$ are injective. We can see $M$ as a direct limit of finitely generated modules. Is it true that $\lim\limits_{\longleftarrow_n}\lim\limits_{\longrightarrow_j}Hom(M_n,N_j)$=$\lim\limits_{\longrightarrow_j}\lim\limits_{\longleftarrow_n}Hom(M_n,N_j)$? If this is true then we can drop the hypothesis $M$ finitely generated in the answer of Matt E.
EDIT: what about $\mathrm{Hom}(\lim\limits_{\longleftarrow}\;M_n,N)$?