Uniqueness of a homomorphism in a projective limit dynamical system; showing that a projective limit system is compact - Is my proof correct?

Solution 1:

Your solution to 1) seems to be essentially correct.

Regarding 2), your interpretation 2ii) is the correct one. More specifically, there may be many homomorphisms from an anonymous $(L,\psi)$ to the inverse limit system $(K,\phi)$, but a $\sigma_\bullet:(L,\psi)\to (K_\bullet,\phi_\bullet)$ ($\bullet$ signifies the whole family) determines a unique homomorphism $\tau: (L,\psi)\to(K,\phi)$ (which might as well be denoted by $\sigma$ according to the conventions of notation of the exercise in question). As you said if $\sigma_\bullet : L\to K_\bullet$ exists, since $K$ is defined as a certain subset of $\prod K_\bullet$, there is a unique arrow $\tau : L\to \prod K_\bullet$, whose coordinates are given by $\sigma_\bullet$. If in addition $\sigma_\bullet$ is compatible with the projections $\pi_{\bullet\bullet}$, then the image of $\tau$ is a subset of $K$, and the fact that $\sigma_\bullet$ is a homomorphism $\psi\to \phi_\bullet$ gives that it's a homomorphism $\psi\to \phi$.

Conversely, if $\tau:(L,\psi)\to (K,\phi)$ is an anonymous homomorphism, then we have $\sigma_\bullet:=\pi_\bullet\circ \tau: (L,\psi)\to (K_\bullet,\phi_\bullet)$ compatible with $\pi_{\bullet\bullet}$ and based on the previous paragraph if $\tau': (L,\psi)\to (K,\phi)$ is an anonymous homomorphism with $\pi_\bullet\circ \tau'=\pi_\bullet\circ \tau$, then $\tau'=\tau$. For practical purposes all this means that when dealing with homomorphisms to/from the inverse limit system you may do so component-wise.

Regarding your use of "ridiculous", the point of category theory (when applicable) is to showcase the non-arbitrariness of constructions that might seem otherwise (see e.g. "The purpose of being categorical is to make that which is formal, formally formal" what does it mean?).