Even if the answer is no, I am interested in a more specific question.


Let $\Sigma$ be a set of operations of finite arity, $E$ be a set of equations over $\Sigma$ and $\mathcal{A}(\Sigma,E)$ be the respective category of algebras and algebra morphisms. Also denote the free algebra functor by $F: \mathsf{Set} \to \mathcal{A}(\Sigma,E)$.


If $f : A \to B$ is a monomorphism in such a category i.e. an injective algebra morphism, and also $X$ is set, does it follow that $FX + f : FX + A \to FX + B$ is also injective?


Any help much appreciated.



Do coproducts in categories of algebras preserve monos?
Well, I have a counterexample.


Let $\Sigma$ contain two unary operations $a$ and $b$. 


Further suppose $E$ contains the equations $a(p) = a(q)$ and $b(p) = b(q)$.


Then $F0 = \emptyset$ because $\Sigma$ contains no nullary operations and we also have the terminal algebra $\mathsf{1}$ with singleton carrier $\{\*\}$. Furthermore the free algebra on omega generators $F\omega$ has carrier $\omega + 1 + 1$, the omega corresponding to the generators, the remaining two elements being the equivalence class containing $a$-prefixed terms and the equivalence class containing $b$-prefixed terms.


Now we certainly have the injective algebra morphism $\iota : \emptyset \hookrightarrow \mathsf{1}$.


However $F\omega + \emptyset \cong F\omega$ so it has carrier $\omega + 1 + 1$, whereas $F\omega + \mathsf{1}$ has carrier $\omega + 1$. This follows because in $F\omega + \mathsf{1}$ we can deduce: 
\[
a(x) = a(\*) = \* = b(\*) = b(x)
\]
for any $x \in U(F\omega + \mathsf{1})$. 


Since $F\omega + \iota$ is surjective but not injective, the respective functor $F\omega + Id$ does not preserves monos.


