Contractible iff every map $f :X \to Y$, for arbitrary $Y$, is nullhomotopic.  I tend to write very inaccurate arguments even when my proof is correct. So I am wondering if someone would be willing to help me take a look at this proof? Thank you very much!

(1) Show that a space $X$ is contractible iff every map $f :X \to Y$, for arbitrary $Y$, is nullhomotopic.

First we show that if a space $X$ is contractible, then every map $f :X \to Y$, for arbitrary $Y$, is nullhomotopic.
That is given a family of maps such that $f_0(X) = X, f_1(X) = x, $  \forall t: f_t(x) = x,  and we want to show that exist a family of maps such that $\hat{f}_0(X) = f(X), \hat{f}_1(X) = y$, for some fixed $y \in Y$.
We let
$$\hat{f} = f \circ f_t,$$
and this is our desired $\hat{f}_t$ because it satisfies the criterion listed above.

Now we show that if every map $f :X \to Y$, for arbitrary $Y$, is nullhomotopic, then $X$ is contractible.
We pick $Y = X$ and $f = \mathbb{I}$, hence $f :X \to X$ is nullhomotopic means the identity map on $X$, $\mathbb{I}|_X$ is homotopic to a constant map. That is to say $X$ is contractible.

(2) Show $X$ is contractible iff every map $f : Y \to X$ is nullhomotopic.

First we show that if $X$ is contractible, then every map $f : Y \to X$ is nullhomotopic.
Given $X$ is contractible, we know that $\mathbb{I} \cong c$ by the homotopy $f_t$. Then for every $f: Y \to X$, we know $f \cong c^\prime$ by the homotopy $\hat{f}_t = f_t \circ f$.

Now we show that if every map $f : Y \to X$ is nullhomotopic, then $X$ is contractible.
Since every map $f : Y \to X$ is nullhomotopic, we pick $f = \mathbb{I}$, and hence $Y$ need to be $X$. So now we have the map $\mathbb{I}: X \to X$ that is homotopy to a constant map. This is to say, $X$ is contractible.
 A: If I look at this, I get t If one's proofs are easier to read and manage to clearly show the method, one can often get away with inaccuracies, whereas they'll always be noted if one has to figure out the meaning of every word or symbol.he feeling that the main problem might actually be style. I'll at the moment only look at the first part and it is clear from it that you know what's going on. However, you seem to have a hard time conveying this to the person who has to read it (I guess some teaching assistant). I don't mean to be degrading or insulting and hope you can somehow use what I'm writing. An easy bit of advise would be to read your own proofs when you're done, as if you didn't know the solution yet. Would you still understand what happens?
More concretely, what would help the first part? First of all, if there are two 'kinds of maps' involved, just give them different names. When there are maps $X\to Y$ and $\to X$, which is the case right now, don't call all of them $f$ with some added decorations. Once you need lots of maps, you'll have to reuse the same letter sometimes, but you could easily add a letter in this case. Then, where does the first sentence ('First we show that...') come from? Contractibility of $X$! Just say so and the TA doesn't need to think about this anymore. Also, do you really just want $f_0(X)=X$ instead of $f_0=\text{id}_x$? (and again, later on, you want $f=\hat{f}_0$ - not just the same image)
This may well be an illustration of what I wrote before. It is an error and at the moment very likely to go noticed - even when your thoughts are clearly correct. Would you have written the following containing the same error, it might just pass a TA without him/her noticing: "Since the space $X$ is contractible, there exists a continuous family of maps $(f_t)$ with $t\in[0,1]$, such that $f_0(X)=X$ and $f_1(X)=x$.
I think the biggest inaccuracy is gone now. Just one small thing: once you define $\hat{f}$, you first lose the subscript $t$ and then all of a sudden it's there again. This should be there all the time. Also not wrong but making it harder for the reader: if $\mathbb{I}$ is a map from $X$ to $X$, you can just as well omit the $|_X$.
