  We give a general, direct and explicit construction of lower-bounded
generalized twisted modules satisfying a universal property for a
grading-restricted vertex (super)algebra $V$ associated to an automorphism $g$
of $V$. In particular, when $g$ is the identity, we obtain lower-bounded
generalized $V$-modules satisfying a universal property. Let $W$ be a
lower-bounded graded vector space equipped with a set of "generating twisted
fields" and a set of "generator twist fields" satisfying a weak commutativity
for generating twisted fields, a generalized weak commutativity for one
generating twisted field and one generator twist field and some other
properties that are relatively easy to verify. We first prove the convergence
and commutativity of products of an arbitrary number of generating twisted
fields, one twist generator field and an arbitrary number of generating fields
for $V$. Then using the convergence and commutativity, we define a twisted
vertex operator map for $W$ and prove that $W$ equipped with this twisted
vertex operator map is a lower-bounded generalized $g$-twisted $V$-module.
Using this result, we give an explicit construction of lower-bounded
generalized $g$-twisted $V$-modules satisfying a universal property starting
from vector spaces graded by weights, $\mathbb{Z}_{2}$-fermion numbers and
$g$-weights (eigenvalues of $g$) and real numbers corresponding to the lower
bounds of the weights of the modules to be constructed. In particular, every
lower-bounded generalized $g$-twisted $V$-module (every lower-bounded
generalized $V$-module when $g$ is the identity) is a quotient of such a
universal lower-bounded generalized $g$-twisted $V$-module (a universal
lower-bounded generalized $V$-module).
