Let $G$ be a group with presentation $$G=\langle x_1,x_2,\cdots,x_k\colon R_1(x_1,\cdots,x_k)=1, \cdots, R_n(x_1,\cdots,x_k)=1\rangle.$$ Here $R_i(x_1,\cdots,x_k)$ denotes a word in $x_1,\cdots,x_k$.
Suppose $y_1,y_2,\cdots,y_k$ are elements of $G$ such that in $G$, $$R_i(y_1,\cdots,y_k)=1.$$
Question: Does the map $x_i\mapsto y_i$ extends to homomorphism (endomorphism) of $G$?
I was thinking YES; but unable to give clear justification. Any suggestion on problem?