We study the category of Reedy diagrams in a
$$\mathscr {V}$$
-model category. Explicitly, we show that if *K* is a small category,
$$\mathscr {V}$$
is a closed symmetric monoidal category and
$$\mathscr {C}$$
is a closed
$$\mathscr {V}$$
-module, then the diagram category
$$\mathscr {V}^K$$
is a closed symmetric monoidal category and the diagram category
$$\mathscr {C}^K$$
is a closed
$$\mathscr {V}^K$$
-module. We then prove that if further *K* is a Reedy category,
$$\mathscr {V}$$
is a monoidal model category and
$$\mathscr {C}$$
is a
$$\mathscr {V}$$
-model category, then with the Reedy model category structures,
$$\mathscr {V}^K$$
is a monoidal model category and
$$\mathscr {C}^K$$
is a
$$\mathscr {V}^K$$
-model category provided that either the unit 1 of
$$\mathscr {V}$$
is cofibrant or
$$\mathscr {V}$$
is cofibrantly generated.