Could you given an example (link) of where this phrase is used to give us some context?
Off the top of my head saying:
"B is the homomorphic image of A"
is equivalent to saying there exists a homomorphism from A to B.
In higher level abstract algebra one may define algebraic object in terms of free objects modulo identities.
Thus for example the complex numbers can be defined as the set of polynomials in the variable i, modulo the identity i2
Another example is the tensor algebra on vector spaces. The tensor product is essentially the free (linear) product.
You can define a free group on say 2 letters by assuming the letters, a, b are generators of a group and all expressions which are not identical (modulo associativity and inverse property) are distinct.
Hence the free group generated by a and b contain distinct elements defined by all products of powers (+ and -) of a and b. Example: a-2
No rearrangement is possible without some identity relating ab to ba.
You can then invoke an identity such as ab = b-1
a to define a smaller group.
Why do it this way? Well when comparing groups it is often easier to show the identities defining them are equivalent than to directly construct an isomorphism. Also if one set of identities implies another set (but not vis versa) that tells us there is a homomorphism (in a direction dual to implication). In general one can study algebraic objects by studying the identities. There is a category relationship and a functor between sets of identities and the objects they define from free objects.