Using both should lead in spiral fashion to both better formal systems and more knowledge about that which they describe. In this spirit we should surely expose the defects of a formal system rather than conceal them. The formal system may need to be changed. The objective necessity that every map has both a domain and a codomain is one of the great principles of category theory. Already in the 30s it was known to be required by even the simplest functors such as pi-zero.
All areas of mathematics lead to qualitative leaps like the functors of algebraic topology.
We carry the inclusions around only as long as we need them. Of course some improved version of tt is needed to handle the presentation of cccs, but the basic issue of how to deal formally with inclusions and memberships between subsets arising as equalizers arises already with Lex. In general given any left adjoint, such as the one from Cat to Lex or from Cat to CCC or the one from sets to monoids,etc, there is an objective notion of presentation that arises. Formal systems are subjective instruments for describing and clarifying such presentations. Naturally such formal systems would involve specifying both domains and codomains.