Consider a $n$-dimensional random vector $\boldsymbol{X}$ with covariance matrix $\mathbf{\Sigma} = (\sigma_{ij})$. We may apply a element-wise scaling to $\boldsymbol{X}$ by multiplying it with a diagonal matrix $\mathbf{A} = \operatorname{diag}(a_1, \dots, a_n)$. Then the covariance matrix of the scaled random vector $\mathbf{A} \boldsymbol{X}$ is $$ \operatorname{Cov}(\mathbf{A} \boldsymbol{X}) = \mathbf{A} \boldsymbol{X} \mathbf{A}^\top = (a_i a_j \sigma_{ij}). $$
My question is: Does this result generalize to stochastic processes on $\mathbb{R}$? In other words: Considering a stochastic process $X(t)$ with covariance function $c_X(t, t')$ and a (non-random) function $f(t)$, what is the covariance function of the stochastic process $Y(t) = f(t) X(t)$? Is it $c_Y(t, t') = f(t) f(t') c_X(t, t')$ or are things more complicated?
I am not very familiar with rigorous probability theory, but I appreciate any hints on the question or suggestions on where to read up on the topic. Thank you very much!