First order structures: relational vs algebraic and their behaviour with respect to morphisms.

Question

I am beginning to study first order logic and noticed that we can have two special interesting cases.

1. Relational structures, which are intepretations of languages with only relational symbol;
2. Algebraic structures, which are intepretations of languages with only functional symbols;

This two structures seem to have "dual" behaviour with respect to morphisms, in the sense that:

• Relational structures are well behaved with respect to the formation of substructures: any nonempty set automatically determines one. Intuitively then their structure 'propagates' inward.
• Algebraic structures are well behaved with respect to the formation of morpshisms: any bijective morphism among two of them is automatically an isomorphism. Intuitively then their structure 'propagates' outward.

I was wondering if this kind of symmetric behaviour extends to other facts and which are other interesting, if any, comparison between these two kinds of structures.