Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces.
There are forgetful functors $$ U_{\mathbf{LRS}}: \mathbf{LRS} \to \mathbf{RS} $$ (the inclusion of the (non-full) subcategory) and $$ U_{\mathbf{RS}}: \mathbf{RS} \to \mathbf{Top} $$ (forgetting the structure sheaf). All three categories have all (small) limits and colimits.
- Are $U_{\mathbf{LRS}}$ and $U_{\mathbf{RS}}$ (right?) adjoint functors?
- Do $U_{\mathbf{LRS}}$ and $U_{\mathbf{RS}}$ perserve pushouts?
I am interested in these questions because I can think easier of topological spaces than of ringed (or locally ringed) spaces. For example, when I intuitively want to see what the pushout of two (locally) ringed spaces is, I want to see first what happens on topological spaces and afterwards think of what is going on with the structure sheaves. Am I allowed to do this?