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. cofaces), so if there's a simpler answer in that case I'd be very happy to know it.

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.

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 $K=\Delta$, so if there's a simpler answer in that case I'd be very happy to know it.

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. cofaces), so if there's a simpler answer in that case I'd be very happy to know it.