  We consider necessary and sufficient conditions for finite generation and
finite presentability for fiber products of free semigroups and free monoids.
We give a necessary and sufficient condition on finite fiber quotients for a
fiber product of two free monoids to be finitely generated, and show that all
such fiber products are also finitely presented. By way of contrast, we show
that fiber products of free semigroups over finite fiber quotients are never
finitely generated. We then consider fiber products of free semigroups over
infinite semigroups, and show that for such a fiber product to be finitely
generated, the quotient must be infinite but finitely generated,
idempotent-free, and $\mathcal{J}$-trivial. Finally, we construct automata
accepting the indecomposable elements of the fiber product of two free
monoids/semigroups over free monoid/semigroup fibers, and give a necessary and
sufficient condition for such a product to be finitely generated.
