How does coalition limit the PM?

Under normal circumstances, the convention of collective responsibility can strengthen the prime minister, as they can rely upon government ministers to support government policy in public.
However, due to coalition government, David Cameron has to work alongside a Deputy Prime Minister from a different party, negotiating to ensure that he has the support of the Liberal Democrats.
LDs mostly supported coalition policies in public but in 2013 Clegg publicly contradicted the work of the Education Sec on academies and free schools.
It is one thing for the PM to face criticism from the opposition, or the backbenches, but it was seen by many commentators as much more significant when coming from a Cabinet Minister.

The prime minister has a number of institutional powers that make the position significantly more powerful than other ministerial positions. For example, the power of patronage – the ability to hire and fire members of the cabinet and government – is an obvious and significant privilege.
However, as a coalition, this power has been constrained, as Cameron is not free to choose Conservative MPs for all positions of government.
Instead, Deputy Prime Minister Nick Clegg has the main influence over the appointment of Liberal Democrat ministers for 5 of the 23 cabinet ministerial posts and 20 out of the 102 remaining government posts. In addition, Cameron has no power to change Clegg’s role or to reshuffle him into a different ministerial position.
It is vital that prime ministers are able to inspire strong support from their MPs, in order to pass the legislation, they are aiming for. This is obviously a far greater challenge as part of a coalition, as Cameron needs to not only influence members of his own party, but possibly many Liberal Democrat MPs as well.

Cameron certainly didn’t have to worry about Nick Clegg challenging his position as leader of the Conservative Party.
Clegg may be making public criticisms, but as the leader of another party, this perhaps weakens the impact these comments make on Cameron’s position.
Clegg has often acted as a ‘human shield’ for Cameron, attracting such negative press that any mistakes made by the Prime Minister have seemed less negative.
Clegg’s early willingness to support the government’s plans to raise university tuition fees drew a terrible backlash in the media and saw his personal popularity plummet.
While the presence of a Deputy Prime Minister might appear to challenge the authority of the Prime Minister, the position can actually strengthen prime ministerial power under the right circumstances.