Perfect secrecy of the reverse of a crypto system that has perfect secrecy

I am trying to solve a problem that reads as follows:

Let $E_1 = (\text{Gen}_1, \text{Enc}_1, \text{Dec}_1)$ be a crypto system that has perfect secrecy. Denote the message space $\mathbb M_1$, the key space $\mathbb K_1$ and the cyphertext space $\mathbb C_1$ ($\mathbb M_1=\mathbb C_1 = \mathbb T, \mathbb K_1 = \mathbb K$). Let $E_2 = (\text{Gen}_2, \text{Enc}_2, \text{Dec}_2)$ be a crypto system with the same message space, cyphertext space and key space as in $E_1$, with the only change that $\text{Enc}_2(k,m)=\text{Dec}_1(k,m)$. Does $E_2$ also have perfect secrecy?

What I tried to do is the following:

By definition of perfect secrecy we have that for any distribution $D_1$ over $\mathbb M_1= \mathbb T$ and for any $(m,c)\in \mathbb T \times \mathbb T$ with $\Pr[C_1 = c | M_1 = m]= Pr[C_1 = c]$ where $Pr[M_1 = m] > 0$ ($M_1,C_1$ are random variables).

Now we assume some distribution $D_2$ over $\mathbb M_2=\mathbb T$, and try to prove that this cryptosystem has perfect secrecy like so: Let $m\in \mathbb M_2 = \mathbb T$ such that $Pr[M_2 = m] > 0$, and let $c\in \mathbb C_2 = \mathbb T$, then:

$Pr[C_2 = c| M_2 = m] = Pr[\text{Enc}_2(K,m)=c] = Pr[\text{Dec}_1(K,m)=c] = $

Here there were two ways of going forward. $K$ is a random variable that is distributed by some distribution over $\mathbb K$ and that has to be similar for both crypto systems because they use the same $\text{Gen}$ algorithm. So:

$Pr[\text{Dec}_1(K,m)=c] = \sum_{k:\text{Dec}_1(k,m)=c} Pr[K=k]$ but I don't know what the decoding algorithm looks like so no idea of what these probabilities might be.

Another approach was to assume (perhaps falsely) that $Pr[\text{Dec}_1(K,m)=c] = Pr[M_1 = c | C_1 = m]$ and given the perfect secrecy of $E_1$ that works out to be equal to $Pr[M_1 = c]$

And $Pr[C_2 = c] = \sum_{m\in\mathbb M_2} Pr[M_2 = m]*Pr[C_2 = c | M_2 = m] = \sum_{m\in\mathbb M_2} Pr[M_2 = m]*Pr[M_1 = c] = Pr[M_1 = c]*\sum_{m\in\mathbb M_2} Pr[M_2 = m] = Pr[M_1 = c]*1 = Pr[M_1 = c]$

And so they are equal, therefore $E_2$ has perfect secrecy.

My concern is that the assumption I made before was false, and that $Pr[C_1 = c] could be equal to 0, in which case this is wrong too.

I feel like I'm lacking in understanding what perfect secrecy really means and so I only have this definition in hand to work with..

Any ideas if this is correct?

Thanks in advance.