  We study the subcategory of topological operads $P$ such that $P(0) = *$ (the
category of unitary operads in our terminology). We use that this category
inherits a model structure, like the category of all operads in topological
spaces, and that the embedding functor of this subcategory of unitary operads
into the category of all operads admits a left Quillen adjoint. We prove that
the derived functor of this left Quillen adjoint functor induces a left inverse
of the derived functor of our category embedding at the homotopy category
level. We deduce from this result that the derived mapping spaces associated to
our model category of unitary operads are homotopy equivalent to the standard
derived operad mapping spaces, which we form in the model category of all
operads in topological spaces. We prove that analogous statements hold for the
subcategory of $k$-truncated unitary operads within the model category of all
$k$-truncated operads, for any fixed arity bound $k\geq 1$, where a
$k$-truncated operad denotes an operad that is defined up to arity $k$.
