I'm trying to make commutative diagrams to go with my math notes.  

So far, I've experimented with AMScd and TikZ-cd. But here's the problem: I need to align the arrows so that I can control where their tails and heads go. For instance, I have the following diagram:

[![enter image description here](https://i.stack.imgur.com/izkmE.png)](https://i.stack.imgur.com/izkmE.png)

I need to align the ϕ arrow's tail to be directly below *U* and its head to be directly above ϕ(*U*). Similarly, for the ψ arrow.

Ideally, I'm looking for something simple (like the ampersand in the matrix, align, or eqnarray environments) because I'll need to make such diagrams throughout the document. Also, there will be more complicated diagrams with diagonal arrows!

[![enter image description here](https://i.stack.imgur.com/ZSycK.png)](https://i.stack.imgur.com/ZSycK.png)

I've experimented so far, unsuccessfully, with AMScd and TikZ-cd. I'm not really familiar with the TikZ package -- so I haven't tried using its matrix, or graph, or whatever it is. Here is the AMScd and TikZ-cd code that generatse the first diagram (I'm not really worried about aligning the diagonal arrow): 

Using AMScd: 

```
\[
\begin{CD}
x \in U \subseteq M @>f>> f\left(x\right) \in V \subseteq N \\
@V\phi VV @VV\psi V \\
\phi\left(x\right) \in \phi\left(U\right) \subseteq \mathbb{R}^{m} @>>\psi f \phi^{-1}> \psi\left(f\left(x\right)\right) \in \psi\left(V\right) \subseteq \mathbb{R}^{n}
\end{CD}
\]

```

Using TikZ-cd:

```
\[
\begin{tikzcd}
x \in U \subseteq M \arrow{r}{f}
\arrow{d}{\phi} &
f\left(x\right) \in V \subseteq N
\arrow{d}{\psi} \\
\phi\left(x\right) \in \phi\left(U\right) \subseteq \mathbb{R}^{m}
\arrow{r}{\psi f \phi^{-1}} &
\psi\left(f\left(x\right)\right) \in \psi\left(V\right) \subseteq \mathbb{R}^{n}
\end{tikzcd}
\]

```

Any help would be welcome. Thanks in advance!