Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions:


1. If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a monomorphism in $\mathcal{C}$ at every component, is it a monomorphism in $Fun(K,\mathcal{C})$?
2. If every 1-simplex in the image of $f$ is a monomorphism, is it true that the maps $lim(f)\to f(\sigma)$ are monomorphisms for all 0-simplices $\sigma\in K$?


I don't know if these are going to be true at this level of generality. Ultimately I'm interested in the very special case of $\mathcal{C}=Gpd\_\infty$ and $K$ being the subcategory of $\Delta$ spanned by injections (i.e. semicosimplicial objects), so if there's a simpler answer in that case I'd be very happy to know it.



Monomorphisms of diagrams in an $\infty$-category
For completeness, and because I cannot figure out the general case (cf. my comment below Daniel's answer), let me prove the following: if $f:\Delta\to C$ is a functor which, when restricted to $\Delta\_{inj}$ takes values in monomorphisms, then $\lim\_\Delta f\to f([0])$ is a monomorphism.


This is simply the following observation : $f([0])\simeq \lim\_\Delta f(-\*[0])$ (as $f(-\*[0])$ is a split-augmented simplicial object), and along thsi identification, the projection map corresponds to $\lim\_\Delta (f\to f(-\*[0]))$. Now each $[n]\to [n]\*[0]$ is an injection, and so this map is $\lim\_\Delta$ of monomorphisms, and hence is a monomophism itself.


This seems to use something specific about the combinatorics of $\Delta$.


When $K$ is general, and say has a final object $\infty$, then one can argue similarly: one has a composite $\lim\_K f\to f(k) \to f(\infty)$, and thus by some version of 2-out-of-3 it suffices to prove that $\lim\_K f\to f(\infty)$ is a monomorphism, but this follows from it simply being $\lim\_K (f\to f(\infty))$. My attempt for a general $K$ with $|K|\simeq \*$ was to try and prove that the inclusion $K\to K^\triangleright$, where we add a final object, preserves the property of every map being a mono, i.e. that for every $k$, $f(k)\to \mathrm{colim}\_K f$ would be a monomorphism - this step is where one would use $|K|\simeq \*$, but I didn't manage to prove it.


There is a somewhat easy-to-prove version which is when $K$ is weakly contractible via a zig-zag of functors : if there is a zigzag $id\_K \leftarrow h\_1 \to h\_2 \leftarrow \dots \to h\_n$ where $h\_n$ is a constant functor, then one can do a similar argument to the one for $\Delta$ and prove the result. I don't remember exactly how often we can expect this: is it the case that any weakly contractible $K$ is a filtered colimit of $K$'s admitting such a zigzag ?


