TITLE: Does "finitely presented" mean "always finitely presented", considered in general
QUESTION [9 upvotes]: I'm wondering about the question

"If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" 

I know this is true for groups and for $R$-modules.  Does anyone know whether this is true for $A$-algebras?  Commutative $A$-algebras?  Other things people might happen to know about it for?

REPLY [9 votes]: Yes in general.
See  Adámek and Rosicky, Locally Presentable and Accessible Categories Cambridge University Press, Cambridge, (1994).
T. 3.12 p. 143.
Of course "in general" I mean: every "algebraic theory" (many sorted) Set models. 
For topological algebraic structures, (like profinite groups) this equivalence isn't true for the  Yiftach Barnea example