  For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to
\mathrm{Aut} \;(G, \cdot),~~a \mapsto \lambda_a$, where $\lambda_a(b) = a^{-1}
\cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then
$\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\mathrm{Aut}\; (G,
\cdot)$, which, in general, may not be a homomorphism. We study skew left
braces $(G, \cdot, \circ)$ for which $\lambda : (G, \cdot) \to \mathrm{Aut}\;
(G, \cdot)$ is a homomorphism. Such skew left braces will be called
$\lambda$-homomorphic. We formulate necessary and sufficient conditions under
which a given homomorphism $\lambda : (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$
gives rise to a skew left brace, which, indeed, is $\lambda$-homomorphic. As an
application, we construct skew left braces when $(G, \cdot)$ is either a free
group or a free abelian group. We prove that any $\lambda$-homomorphic skew
left brace is an extension of a trivial skew brace by a trivial skew brace.
Special emphasis is given on $\lambda$-homomorphic skew left brace for which
the image of $\lambda$ is cyclic. A complete characterization of such skew left
braces on the free abelian group of rank two is obtained.
