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