When you click Check In button, method handleCommand called from javascript object SP.Ribbon.PageState.PageStateHandler (sp.ribbon.js) with commandId = "PageStateGroupCheckin"
If you debug it, you can find that it executes this line at least:
return SP.Ribbon.PageState.Handlers.showStateChangeDialog(properties['CommandValueId'], SP.Ribbon.PageState.ImportedNativeData.CommandHandlers[properties['CommandValueId']]);
EDIT
Ok, somebody down voted, I'll explain how to find this function.
First of all inspect check in button using firebug or google chrome dev tools. Check in button is a link with id Ribbon.WikiPageTab.EditAndCheckout.Checkout.Menu.Checkout.Checkin-Menu16. Next, go to cmdui.xml file in 14 hive. Here using search find button with id Ribbon.WikiPageTab.EditAndCheckout.Checkout.Menu.Checkout. You will find this menu:
<Menu Id="Ribbon.WikiPageTab.EditAndCheckout.Checkout.Menu">
and button with definition:
<Button
Id="Ribbon.WikiPageTab.EditAndCheckout.Checkout.Menu.Checkout.Checkin"
MenuItemId="Ribbon.WikiPageTab.EditAndCheckout.Checkout.Menu.Checkout.Checkin"
Sequence="20"
Alt="$Resources:core,ButCheckin;"
Command="PageStateGroupCheckin"
CommandValueId="PageStateGroupCheckin"
Image16by16="/_layouts/$Resources:core,Language;/images/formatmap16x16.png" Image16by16Top="-104" Image16by16Left="-32"
Image32by32="/_layouts/$Resources:core,Language;/images/formatmap32x32.png" Image32by32Top="-416" Image32by32Left="-192"
LabelText="$Resources:core,ButCheckin;"
ToolTipTitle="$Resources:core,ButCheckin;"
ToolTipDescription="$Resources:core,cui_STT_ButCheckin_page;"
/>
We interested in CommandValueId attribute. Next go to sp.ribbon.debug.js and using search find object, which handle this particular commandid - this is SP.Ribbon.PageState.PageStateHandler. All commands in page component handle method handleCommand, setup break and click check in in browser - consider screen shot below:

if you step further you got to line, that I mentioned above:
It means, that when you press Check In in the ribbon method SP.Ribbon.PageState.Handlers.showStateChangeDialog fires.
Hope this helps.