Definition: A continuous lattice $D$ is said to be a projection of a continuous lattice $D'$ if and only if there are a pair of continuous maps $$i:D\rightarrow D'$$ and $$j:D'\rightarrow D$$ such that $$j\circ i = \operatorname{id}_D\tag{1}$$ and $$i\circ j \sqsubseteq \operatorname{id}_{D'}\tag{2}$$ Where $\sqsubseteq$ is the ordering on the function space of all functions to $D'$.
This property seems to be very useful and can be shown to have very interesting consequences. More on this in this link. I was wondering, whether this property could be generalized to category of groups, rings, modules... or more generally, to any category. Surely condition (1) can be said everywhere, for partially ordered sets (2) could be said. Any idea how to extend this further?
There's no natural way to generalize condition (2) to an arbitrary category, but it's clear whenever the homs are ordered, as mentioned in the comments. More generally in a 2-category one can speak of adjoint retracts, in which still $j\circ i$ is the identity of $D$ but now we are given a natural map $i\circ j\to\mathrm{id}_C$, which is trivial on the image of $i$ and under $j$. These are a stricter version of reflective subobjects, which are constantly useful, especially in the case of reflective subcategories.